PERSPECTIVES ON SPIN GLASSES

Transcription

1 PERSPECTIVES ON SPIN GLASSES Presenting and developing the theory of spin glasses as a prototype for complex systems, this book is a rigorous and up-to-date introduction to their properties. The book combines a mathematical description with a physical insight of spin glass models. Topics covered include the physical origins of those models and their treatment with replica theory; mathematical properties such as correlation inequalities and their use in the thermodynamic limit theory; main exact solutions of the mean field models and their probabilistic structures; and the theory of the structural properties of the spin glass phase such as stochastic stability and the overlap identities. Finally, a detailed account is given of the recent numerical simulation results and properties, including overlap equivalence, ultrametricity, and decay of correlations. The book is ideal for mathematical physicists and probabilists working in disordered systems. pierluigi contucci is Professor of Mathematical Physics at the University of Bologna, and Research Director for the hard sciences section of the Istituto Cattaneo. His research interests are in statistical mechanics and its applications to socio-economic sciences. cristian giardinà is Associate Professor in Mathematical Physics at the University of Modena and Reggio Emilia, and Visiting Professor in Probability at Nijmegen University. His research interests are in mathematical statistical physics and stochastic processes. in this web service

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3 PERSPECTIVES ON SPIN GLASSES PIERLUIGI CONTUCCI University of Bologna CRISTIAN GIARDINÀ University of Modena and Reggio Emilia in this web service

4 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by, New York Information on this title: / C P. Contucci and C. Giardinà 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of. First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Contucci, Pierluigi, 1964 Perspectives on spin glasses /. pages cm Includes bibliographical references and index. ISBN Spin glasses Mathematical models. I. Giardinà, Cristian. II. Title. QC176.8.S68C dc ISBN Hardback has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. in this web service

5 Contents Preface page vii 1 Origins, models and motivations The spin glass problem Random interactions, finite-dimensional models, mean-field models Quenched measure and real replicas Definition of a mean-field spin glass Replica method for the SK model The replica symmetric solution The ultrametric replica symmetry breaking solution 19 2 Correlation inequalities Spin functions Ferromagnetism and the Griffiths Kelly Sherman inequalities Spin glass correlation inequalities of type I Extension to quantum case Type II inequalities: results and counterexamples Inequalities of types I and II on the Nishimori line Some consequences of correlation inequalities 44 3 The infinite-volume limit Introduction Finite-dimensional models with Gaussian interactions Pressure self-averaging Finite-dimensional models with centered interactions Quantum models Extension to non-centered interactions Independence of the pressure from boundary conditions 64 v in this web service

6 vi Contents 3.8 Surface pressure Complete theory on the Nishimori line Thermodynamical limit for mean-field models, REM and GREM 78 4 Exact results for mean-field models Introduction to the Poisson point process Poisson point process with exponential intensity The Poisson Dirichlet distribution The random energy model Upper bound for the REM pressure Lower bound for the REM pressure Statistics of energy levels for the REM The generalized random energy model Upper bound for the GREM pressure Lower bound for the GREM pressure The Aizenman Sims Starr extended variational principle Guerra upper bound Talagrand theorem and open problems Spin glass identities The stability method and the structural identities Stochastic stability identities Strong stochastic stability and marginal independence Graph theoretical approach to Aizenman Contucci identities Identities from self-averaging Identities on the Nishimori line Interaction flip identities Numerical simulations Introduction Simulations with real replicas Overlap equivalence Ultrametricity Decay of correlations Energy interfaces 192 References 200 Index 209 in this web service

7 Preface Spin glasses are statistical mechanics systems with random interactions. The alternating sign of those interactions generates a complex physical behavior whose mathematical structure is still largely uncovered. The approach we follow in this book is that of mathematical physics, aiming at the rigorous derivation of their properties with the help of physical insight. The book starts with the theoretical physics origins of the spin glass problem. The main models are introduced and a description of the replica approach is illustrated for the Sherrington Kirkpatrick model. Chapters 2 and 3 contain the starting points of the mathematical rigorous approach leading to the control of the thermodynamic limit for spin glass systems. Correlation inequalities are introduced and proved in various settings, including the Nishimori line. They are then used to prove the existence of the large-volume limit in both short-range and mean-field models. Chapter 4 deals with exact results which belong to the mean-field case. The methods and techniques illustrated span from the Ruelle probability cascades to the Aizenman Sims Starr variational principle. In this framework the Guerra upper bound theorem for the pressure is presented and the Talagrand theorem is reported. Chapter 5 deals with the structural identities characterizing the spin glass phase. These are obtained by an extension of the stochastic stability method, i.e. an invariance property of the system under small perturbations, together with the self-averaging property. Chapter 6 features some problems which are still out of analytical reach and are investigated with numerical methods: the equivalence among different overlap structures, the hierarchical organization of the states, the decay of correlations, and the energy interface cost. Needless to say there are innumerable important issues not covered by the book. Among them are the dynamical properties of spin glasses (see, for example, Sompolinsky and Zippelius (1982); Cugliandolo and Kurchan (1993); Bouchaud vii in this web service

8 viii Preface (1992); Ben Arous et al. (2001, 2002); Bovier et al. (2001)). Another very large topic not covered is that of applications whose ideas originated within spin glass theory and successfully fertilized other areas. It is a pleasure to thank the co-authors whose research work provided the foundations of this book: Michael Aizenman, Alessandra Bianchi, Mirko Degli Esposti, Claudio Giberti, Sandro Graffi, Andreas Knauf, Stefano Isola, Joel Lebowitz, Satoshi Morita, Hidetoshi Nishimori, Giorgio Parisi, Joe Pulé, Shannon Starr, Francesco Unguendoli, and Cecilia Vernia. Useful conversations with many colleagues are acknowledged, particularly those with Louis-Pierre Arguin, Adriano Barra, Anton Bovier, Edouard Brézin, Aernout van Enter, Silvio Franz, Francesco Guerra, Frank den Hollander, Jorge Kurchan, Enzo Marinari, Marc Mezard, Chuck Newman, Dmitry Panchencko, Daniel Stein, and Michael Talagrand. Last but not least we thank Claudio Giberti, Bernardo D Auria, Aernout van Enter, and Cecilia Vernia for their careful reading of the manuscript. in this web service